The authors show that the CUSUM test of the stability over time of the coefficients of a linear regression model, which is usually based on recursive residuals, can also be applied to ordinary least squares residuals. The authors derive the limiting null distribution of the resulting test and compare its local power to that of the standard procedure. It turns out that neither version is uniformly superior to the other.

In this paper we derive the asymptotic distribution of the test statistic of a generalized version of the integrated conditional moment (ICM) test of Bierens (1982, 1984), under a class of Vn-local alternatives, where n is the sample size. The generalized version involved includes neural network tests as a special case, and allows for testing misspecification of dynamic models. It appears that the ICM test has nontrivial local power. Moreover, we show that under the assumption of normal errors the ICM test is asymptotically admissible, in the sense that there does not exist a test that is uniformly more powerful. The asymptotic size of the test is case-dependent: the critical values of the test depend on the data-generating process. In this paper we derive case-independent upperbounds of the critical values

This paper derives asymptotically optimal tests for testing problems in which a nuisance parameter exists under the alternative hypothesis but not under the null. For example, the results apply to tests of structural change with unknown changepoint. The testing problem considered is nonstandard and the classical asymptotic optimality results for the Lagrange multiplier, Wald, and likelihood ratio do not apply. A weighted average power criterion is used here to generate optimal tests. This criterion is similar to that used by A. Wald (1943) to obtain the classical asymptotic optimality properties of Wald tests in 'regular' testing problems.

The well-known CUSUM test for structural change is investigated whe n there are lagged dependent variables among the regressors in a linear model. The authors show that both a modified CUSUM test, suggested b y J. M. Dufour (1982), and the straightforward CUSUM test retain their asymptotic significance levels in dynamic models, and find that the power depends crucially on the angle between the mean regressor and the structural shift.

This paper characterizes empirically achievable limits for time series econometric modeling and forecasting. The approach involves the concept of minimal information loss in time series regression and the paper shows how to derive bounds that delimit the proximity of empirical measures to the true probability measure (the DGP) in models that are of econometric interest. The approach utilizes joint probability measures over the combined space of parameters and observables and the results apply for models with stationary, integrated, and cointegrated data. A theorem due to Rissanen is extended so that it applies directly to probabilities about the relative likelihood (rather than averages), a new way of proving results of the Rissanen type is demonstrated, and the Rissanen theory is extended to nonstationary time series with unit roots, near unit roots, and cointegration of unknown order. The corresponding bound for the minimal information loss in empirical work is shown not to be a constant, in general, but to be proportional to the logarithm of the determinant of the (possibility stochastic) Fisher–information matrix. In fact, the bound that determines proximity to the DGP is generally path dependent, and it depends specifically on the type as well as the number of regressors. For practical purposes, the proximity bound has the asymptotic form (K/2)log n, where K is a new dimensionality factor that depends on the nature of the data as well as the number of parameters in the model. When ‘good’ model selection principles are employed in modeling time series data, we are able to show that our proximity bound quantifies empirical limits even in situations where the models may be incorrectly specified. One of the main implications of the new result is that time trends are more costly than stochastic trends, which are more costly in turn than stationary regressors in achieving proximity to the true density. Thus, in a very real sense and quantifiable manner, the DGP is more elusive when there is nonstationarity in the data. The implications for prediction are explored and a second proximity theorem is given, which provides a bound that measures how close feasible predictors can come to the optimal predictor. Again, the bound has the asymptotic form (K/2)log n, showing that forecasting trends is fundamentally more difficult than forecasting stationary time series, even when the correct form of the model for the trends is known.

This paper develops an asymptotic theory of Bayesian inference for time series. A limiting representation of the Bayesian data density is obtained and shown to be of the same general exponential form for a wide class of likelihoods and prior distributions. Continuous time and discrete time cases are studied. In discrete time, an embedding theorem is given which shows how to embed the exponential density in a continuous time process. From the embedding we obtain a large sample approximation to the model of the data that corresponds to the exponential density. This has the form of discrete observations drawn from a nonlinear stochastic differential equation driven by Brownian motion. No assumptions concerning stationarity or rates of convergence are required in the asymptotics. Some implications for statistical testing are explored and we suggest tests that are based on likelihood ratios (or Bayes factors) of the exponential densities for discriminating between models.

This paper proposes a class of optimal tests for the constancy of parameters in random coefficients models. Our testing procedure covers the class of Hamilton's models, where the parameters vary according to an unobservable Markov chain, but also applies to nonlinear models where the random coefficients need not be Markov. We show that the contiguous alternatives converge to the null hypothesis at a rate that is slower than the standard rate. Therefore, standard approaches do not apply. We use Bartlett-type identities for the construction of the test statistics. This has several desirable properties. First, it only requires estimating the model under the null hypothesis where the parameters are constant. Second, the proposed test is asymptotically optimal in the sense that it maximizes a weighted power function. We derive the asymptotic distribution of our test under the null and local alternatives. Asymptotically valid bootstrap critical values are also proposed.